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Full text of "Treatise On Analysis Vol-Ii"

16    BAIRE'S THEOREM AND ITS CONSEQUENCES       87
coefficients cn e E' are given by

Applying this result to the situation where 7 is a circuit
(Q^t<>2n) in A - {a} we obtain, for any weakly analytic function
on A, the Taylor expansion

(12.16.6.5)                    /,(*) = £ 1 /<»>(*)(Z - a)",

n=0 Wl

convergent in the disk \z — a\ < r (independent ofze E), and the derivatives
are given by

More generally, suppose that in A — {a} we have
(12.16.6.7)                   /z=^_^i_ + ^>

where the x% are linear forms on E (a priori, not necessarily continuous), and
z\~*gz is a weakly analytic function on A — {a} which is weakly bounded in a
neighborhood of a. Then, by (9.15), for each x e E, the function z\-+gs(x) can
be extended by continuity to the point a, and the extended function is analytic
in A. Hence there exists a weakly analytic function zt-*hz in A whose restric-
tion to A - {a} is z\-*gz. Furthermore, the x* are continuous linear forms on
E (i.e., they are elements of E'). Indeed, for each x e E we have

= ^ J(z - a)*

where y is a circuit t\~+a + reu (0 <J tf <i 27c) with r sufficiently small and
independent of x (9.14); our assertion therefore follows from the considera-
tions above (12.16.6.3).

Finally, we remark that the principle of analytic continuation (9.4.2)
remains valid for weakly analytic functions. This is an immediate consequence
of the definitions.

(12.16.7) Let E be a Hilbert space and u an endomorphism of the (non-
topologicaT) vector space E. Then the following three conditions are equivalent:

(a)    u is continuous]

(b)    u is weakly continuous•;

(c)    u has an adjoint (11.5).oints (zn) in A (3.13.13),